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Analysis and control of transitional shear layers using global modes

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Shervin Bagheri Doctoral thesis 2010 Download 12 Mb
ISSN 0348-467X


In this thesis direct numerical simulations are used to investigate two phenomena in shear flows: laminar-turbulent transition over a flat plate and periodic vortex shedding induced by a jet in crossflow. The emphasis is on understanding and controlling the flow dynamics using tools from dynamical systems and control theory. In particular, the global behavior of complex flows is described and low-dimensional models suitable for control design are developed; this is done by decomposing the flow into global modes determined from spectral analysis of various linear operators associated with the Navier--Stokes equations. Two distinct self-sustained global oscillations, associated with the shedding of vortices, are identified from direct numerical simulations of the jet in crossflow. The investigation is split into a linear stability analysis of the steady flow and a nonlinear analysis of the unsteady flow. The eigenmodes of the Navier--Stokes equations, linearized about an unstable steady solution reveal the presence of elliptic, Kelvin-Helmholtz and von K\'arm\'an type instabilities. The unsteady nonlinear dynamics is decomposed into a sequence of Koopman modes, determined from the spectral analysis of the Koopman operator. These modes represent spatial structures with periodic behavior in time. A shear-layer mode and a wall mode are identified, corresponding to high-frequency and low-frequency self-sustained oscillations in the jet in crossflow, respectively. The knowledge of global modes is also useful for transition control, where the objective is to reduce the growth of small-amplitude disturbances to delay the transition to turbulence. Using a particular basis of global modes, known as balanced modes, low-dimensional models that capture the behavior between actuator and sensor signals in a flat-plate boundary layer are constructed and used to design optimal feedback controllers. It is shown that by using control theory in combination with sensing/actuation in small, localized, regions near the rigid wall, the energy of disturbances may be reduced by an order of magnitude.